TY - JOUR

T1 - The circle problem of gauss and the divisor problem of dirichlet—still unsolved

AU - Berndt, Bruce C.

AU - Kim, Sun

AU - Zaharescu, Alexandru

N1 - Publisher Copyright:
© The Mathematical Association of America.

PY - 2018

Y1 - 2018

N2 - Let r2(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for Σn≤x r2(n) and Σn≤x d(n), respectively, as x tends to infinity. But what is the error made in such approximations? Number theorists have been attempting to answer these two questions for over one and one-half centuries, and although we think that we essentially “know” what these errors are, progress in proving these conjectures has been agonizingly slow. Ramanujan had a keen interest in these problems, and although, to the best of our knowledge, he did not establish any bounds for the error terms, he did give us identities that have been used to derive bounds, and two further identities that might be useful, if we can figure out how to use them. In this paper, we survey what is known about these two famous unsolved problems, with a moderate emphasis on Ramanujan’s contributions.

AB - Let r2(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for Σn≤x r2(n) and Σn≤x d(n), respectively, as x tends to infinity. But what is the error made in such approximations? Number theorists have been attempting to answer these two questions for over one and one-half centuries, and although we think that we essentially “know” what these errors are, progress in proving these conjectures has been agonizingly slow. Ramanujan had a keen interest in these problems, and although, to the best of our knowledge, he did not establish any bounds for the error terms, he did give us identities that have been used to derive bounds, and two further identities that might be useful, if we can figure out how to use them. In this paper, we survey what is known about these two famous unsolved problems, with a moderate emphasis on Ramanujan’s contributions.

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U2 - 10.1080/00029890.2018.1401853

DO - 10.1080/00029890.2018.1401853

M3 - Article

AN - SCOPUS:85045928617

SN - 0002-9890

VL - 125

SP - 99

EP - 114

JO - American Mathematical Monthly

JF - American Mathematical Monthly

IS - 2

ER -